\[\sqrt{re \cdot re + im \cdot im}\]
Test:
math.abs on complex
Bits:
128 bits
Bits error versus re
Bits error versus im
Time: 8.2 s
Input Error: 30.3
Output Error: 12.4
Log: ⚲
Profile: 🕒
\(\begin{cases} -re & \text{when } re \le -1.105317861923087 \cdot 10^{+94} \\ \sqrt{{re}^2 + im \cdot im} & \text{when } re \le -2.994294306140617 \cdot 10^{-305} \\ im & \text{when } re \le 2.255503749858442 \cdot 10^{-251} \\ \sqrt{{re}^2 + im \cdot im} & \text{when } re \le 2.3309285477581096 \cdot 10^{+154} \\ re & \text{otherwise} \end{cases}\)

    if re < -1.105317861923087e+94

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      47.5
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      47.5
    3. Applied taylor to get
      \[\sqrt{{re}^2 + im \cdot im} \leadsto -1 \cdot re\]
      0
    4. Taylor expanded around -inf to get
      \[\color{red}{-1 \cdot re} \leadsto \color{blue}{-1 \cdot re}\]
      0
    5. Applied simplify to get
      \[\color{red}{-1 \cdot re} \leadsto \color{blue}{-re}\]
      0

    if -1.105317861923087e+94 < re < -2.994294306140617e-305 or 2.255503749858442e-251 < re < 2.3309285477581096e+154

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      18.9
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      18.9

    if -2.994294306140617e-305 < re < 2.255503749858442e-251

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      44.9
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      44.9
    3. Applied taylor to get
      \[\sqrt{{re}^2 + im \cdot im} \leadsto im\]
      0
    4. Taylor expanded around 0 to get
      \[\color{red}{im} \leadsto \color{blue}{im}\]
      0

    if 2.3309285477581096e+154 < re

    1. Started with
      \[\sqrt{re \cdot re + im \cdot im}\]
      59.6
    2. Applied simplify to get
      \[\color{red}{\sqrt{re \cdot re + im \cdot im}} \leadsto \color{blue}{\sqrt{{re}^2 + im \cdot im}}\]
      59.6
    3. Applied taylor to get
      \[\sqrt{{re}^2 + im \cdot im} \leadsto re\]
      0
    4. Taylor expanded around inf to get
      \[\color{red}{re} \leadsto \color{blue}{re}\]
      0
  1. Removed slow pow expressions

Original test:


(lambda ((re default) (im default))
  #:name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))